Moving Character Along Round Tunnel Walls

What I'm trying to achieve is to align and move character inside a (arbitrary shaped, yet round, twisting and maybe looping) tunnel shape with "feet" always touching the ground. Physics and path following are not options for me, so I started with basic raycast solution:

However the result is not what I expected. At first everything goes smoothly but when character has moved enough to left/right so that he's standing in angle of >90 degrees sideways he starts going through the tunnel wall, eventually going through the wall and preventing raycast from hitting the wall.